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Some properties of locally compact groups

Published online by Cambridge University Press:  09 April 2009

Neil W. Rickert
Neil W. Rickert
Affiliation:
Yale University
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In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 433 - 454
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Chevalley, C., Theory of Lie groups, (Princeton University Press, Princeton, N.J., 1946).Google Scholar
[2]Goto, M., ‘Faithful representations of Lie groups I’, Math. Japonicae, 1 (1948), 107–118.Google Scholar
[3]Goto, M., ‘Faithful representations of Lie groups II’, Nagoya Math. Journal, 1 (1950), 91–107.CrossRefGoogle Scholar
[4]Goto, M., ‘Linear representations of topological groups’, Proc. Amer. Math. Soc., 1 (1950), 425–437.CrossRefGoogle Scholar
[5]Hall, M., The theory of groups, (Macmillan, New York 1959).Google Scholar
[6]Helgason, S., Differential geometry and symmetric spaces, (Academic Press, New York 1962).Google Scholar
[7]Hewitt, E. and Ross, K. A., Abstract harmonic analysis I, (Springer, Berlin and New York 1962).Google Scholar
[8]Iwasawa, K., ‘On some types of topological groups’, Annals of Math., 50 (1949), 507–557.CrossRefGoogle Scholar
[9]Jacobson, N., Lie algebras, (Interscience, New York 1962).Google Scholar
[10]Montgomery, D. and Zippin, L., Topological transformation groups, (Interscience, New York 1955).Google Scholar
[11]Newburgh, J. D., ‘Metrization of finite dimensional groups’, Duke Math. Journal, 20 (1953), 287–293.CrossRefGoogle Scholar
[12]Well, A., L'integration dans les groupes topologiques et ses applications, (Hermann, Paris 1953).Google Scholar
[13]Wang, H., ‘On the deformation of a lattice in a Lie group’, Amer. Journal of Math., 85 (1963), 189–212.CrossRefGoogle Scholar
[14]Zassenhaus, H., ‘Beweis eines Satzes über diskrete Gruppen’, Abh. Math. Seminar, Hamburg, 12 (1938), 289–312.CrossRefGoogle Scholar
[15]Zassenhaus, H., The theory of groups, (Chelsea, New York, 1958).Google Scholar